Abstract: Fourier restriction, Radon-like operators, and decoupling theory are three active areas of harmonic analysis which involve submanifolds of Euclidean space in a fundamental way. In each case, the mapping properties of the objects of study depend in a fundamental way on the "non-flatness" of the submanifold, but with the exception of certain extreme cases (primarily curves and hypersurfaces), it is not clear exactly how to quantify the geometry in an analytically meaningful way. In this talk, I will discuss a series of recent results which shed light on this situation using tools from an unusually broad range of mathematical sources.

Abstract: Various problems in harmonic analysis are intimately connected with studying solutions to additive equations over subsets of curves and surfaces. The latter is amenable to techniques from incidence geometry since we can count such solutions by interpreting them as incidences between points and curves/surfaces. In this talk, we study additive energies of arbitrary subsets of parabolas/convex curves, and their connections to a problem of Bourgain and Demeter regarding a diameter-free version of the quadratic Vinogradov mean value theorem. We also mention some new results associated with additive energies on higher dimensional surfaces which are related to restriction type problems on spheres.